Poisson distribution

The number of times an event occurs in an interval of time, area, volume/space, or set of people.
Probability of k events/arrivals occuring in an interval given λ the expected/mean number of random independent events that occur in an interval.
The sample space S = {0,1,2,...} i.e. the number of events/arrivals; has no upper bound.
P(k) the probability that in the interval there will be k events / arrivals:
P(k) =
AKA The probability mass function.
Alternate notation: let Χ be a random variable, P(Χ=x) =
     x=0,1,...,n

Piscatorial example: suppose λ=4 fish are caught on average in one hour. (There's nothing fishy about Poisson!)
What is the probability of catching k=0 fish, k=1 fish, k=2 fishes, k=3 fishes, etc. in one hour?
Ex.: suppose λ=3 things exist in one cubic meter of some stuff.
What is the probability of observing k=0, 1, 2, 3, 4, 5, 6, etc. things in that cubic meter of stuff?
Ex.: Prussian Army soldiers killed by (accidental) horsekick (mules too?) was 0.61 per year (per cavalry corps).
What is the probability that no soldiers were horsekicked to death in a year, 1 soldier killed, 2, 3, 4 etc.?
Ex.: TikTok clicks occur 5.5 times per microsecond.
What's the chance that more than 7 occur in the next microsecond? ...
Ex.: 1 ng of Pu-239 has an average of 2.3 radioactive decays per second.
What is the probability of a 1 ng sample having k=0,1,2,3,4,... decays in one second?
What is the probability of a 1 ng sample having k=0,1,2,3,4,... decays in 2 seconds? (λ=4.6)

Expected/mean number of events in an interval (λ):   λ∈[0,∞)

mean μ = λ:
standard deviation σ = √λ:        e-λ=

   k        P(k)    ∑P(k)=CDF=P(≤k)   P(≥k)

The table continues up to 2*λ or as long as the probabilty is greater than .001.

Show a particular k:   
exactly k: PMF(k) = P(k) =
at most k (no more than k): CDF(k)=P(≤k) = ∑P(x),x≤k =
at least k (k or more): P(≥k) = 1-CDF(k-1) =

Probability histogram:

When λ=1, i.e. one event per interval, then the probability that no event occurs, P(0), is 1/e≈.37, and the probability that exactly one event occurs, P(1), is also 1/e.

When λ is an integer, P(λ) = P(λ-1)

In 1/m of the interval, the expected/mean number of random independent events that occur in it is λ/m.
In m intervals, the expected/mean number of random independent events that occur in it is mλ.

P(0) = e-λ
P(1) = λe-λ

The Poisson distribution with λ=np approximates the Binomial distribution if n is large and p is small.

As λ gets larger, the pmf functions becomes Normalish:

Excel: Poisson.Dist(k, λ, TRUE/FALSE) FALSE for PMF, TRUE for CDF.


1D example. Events at time:

2D example. Events in area:

V-1 London 576 regions, 535 hits μ=.929 #hits: 0 1 2 3 4 5 #regions:229 211 93 35 7 1